Do you get the same PHP Warning: mysql_query() error in your logs when trying to connect, or is that error now different?
Yes, the same error
[23-Nov-2016 21:02:10 UTC] PHP Warning: mysql_query(): MySQL server has gone away in D:\Hosting\7268586\html\wp-includes\wp-db.php on line 1870
[23-Nov-2016 21:02:10 UTC] PHP Warning: mysql_query(): Error reading result set’s header in D:\Hosting\7268586\html\wp-includes\wp-db.php on line 1870
Alright. This type of error often comes from keeping a mysql connection open without making any requests after a certain time, after which the connection is shut down: http://dev.mysql.com/doc/refman/5.7/en/gone-away.html
Going back to what your hosting provider told you, that time is most likely set to 30 seconds on your server.
There are a few server settings your hosting provider can change to solve that problem; I’d recommend that you get back to them and ask them to take a closer look at the problem, and what they can do to improve the resources available to your site.
Let me know how it goes.
Unfortunately that’s not something they can do on a shared server. We’re looking into other options.